It is quite common to want to vary the resonance frequency of a
resonator in real time. This is a special case of a tunable
filter. In the pre-digital days of analog synthesizers, filter
modules were tuned by means of control voltages, and were thus
called voltage-controlled filters(VCF). In
the digital domain, control voltages are replaced by
time-varying filter coefficients. In the time-varying case,
the choice of filter structure has a profound effect on how the filter
characteristics vary with respect to coefficient variations. In this
section, we will take a look at the time-varying two-pole resonator.

and, since
is real and positive, it coincides with the
amplitude response, i.e.,
.

An important fact we can now see is that the gain at resonance
depends markedly on the resonance frequency. In particular, the
ratio of the two cases just analyzed is

We did not show that resonance gain is maximized at
and minimized at
, but this is straightforward
to show, and strongly suggested by Fig.B.17 (and Fig.B.9).

Note that the ratio of the dc resonance gain to the
resonance
gain is unbounded! The sharper the resonance (the closer
is to 1), the greater the disparity in the gain.

Figure B.17 illustrates a number of resonator frequency responses
for the case
. (Resonators in practice may use values of
even closer to 1 than this--even the case
is used for making
recursive digital sinusoidal oscillators [90].) For
resonator tunings at dc and
, we predict the resonance gain to
be
dB, and this is what we see in the plot.
When the resonance is tuned to
, the gain drops well below 40
dB. Clearly, we will need to compensate this gain variation when
trying to use the two-pole digital resonator as a tunable filter.

Figure B.17:
Frequency response overlays for the
two-pole resonator
, for
and 10 values of
uniformly spaced from 0
to
. The 5th case is plotted using thicker lines.

Figure B.18 shows the same type of plot for the complex
one-pole resonator
, for
and
10 values of
. In this case, we expect the frequency
response evaluated at the center frequency to be
. Thus, the gain at
resonance for the plotted example is
db for all
tunings. Furthermore, for the complex resonator, the resonance gain
is also exactly equal to the peak gain.

Figure B.18:
Frequency response overlays
for the one-pole complex resonator
, for
and 10 values of
uniformly spaced from 0
to
. The 5th case is
plotted using thicker lines.